\(\int 8x^7 \text{ dx} = \dfrac{8}{7 + 1}x^{7 + 1} + \text{ C}\)
\(\int 8x^7 \text{ dx} = \dfrac{8}{8}x^{8} + \text{ C}\)
\(\int 8x^7 \text{ dx} = x^{8} + \text{ C}\)
\(\int \dfrac{2}{5}x^1 \text{ dx} = \dfrac{2}{5}\cdot \dfrac{1}{1 + 1} x^{1 + 1} + \text{ C}\)
\(\int \dfrac{2}{5}x^1 \text{ dx} = \dfrac{\cancel{2}}{5}\cdot \dfrac{1}{\cancel{2}} x^{2} +\text{ C}\)
\(\int \dfrac{2}{5}x^1 \text{ dx} = \dfrac{1}{5}x^{2}+ \text{ C}\)
\(\int \sqrt{x^5} \text{ dx}= \int x^{\frac{5}{2}}\text{ dx}\)
Ingat bahwa \(\color{blue} \sqrt{x} = x^{\frac{1}{2}}\)
\(\int x^{\frac{5}{2}}\text{ dx} = \dfrac{1}{\frac{5}{2} + 1} x^{\frac{5}{2} + 1} + \text{ C}\)
\(\int x^{\frac{5}{2}}\text{ dx} = \dfrac{1}{\frac{7}{2} } x^{\frac{7}{2}} + \text{ C}\)
\(\int x^{\frac{5}{2}}\text{ dx} = \dfrac{2}{7} x^{\frac{7}{2}} + \text{ C}\)
\(\int 100 \text{ dx} = \int 100 x^{0} \text{ dx}\)
\(\int 100 \text{ dx} = 100 \cdot \dfrac{1}{0 + 1} x^{0 + 1} + \text{ C}\)
\(\int 100 \text{ dx} = 100 \cdot \dfrac{1}{1} x^{1} + \text{ C}\)
\(\int 100 \text{ dx} = 100x + \text{ C}\)
\(\int (2x^2 + x \:-\:3) \text{ dx} = \dfrac{2}{2 + 1} x^{2 + 1} + \dfrac{1}{1 + 1} x^{1 + 1} \:-\:3x + \text{ C}\)
\(\int (2x^2 + x \:-\:3) \text{ dx} = \dfrac{2}{3} x^{3} + \dfrac{1}{2} x^{2} \:-\:3x + \text{ C}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \int \dfrac{x^3 \:-\:1}{x^{\frac{1}{2}}} \text{ dx}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \int \left(\dfrac{x^3}{x^{\frac{1}{2}}}\:-\:\dfrac{1}{x^{\frac{1}{2}}} \right)\text{ dx}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \int \left(x^{3\:-\:\frac{1}{2}} \:-\:x^{-\frac{1}{2}}\right) \text{ dx}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \int \left( x^{\frac{5}{2}} \:-\:x^{-\frac{1}{2}}\right) \text{ dx}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \dfrac{1}{\frac{5}{2} + 1} x^{\frac{5}{2} + 1} \:-\:\dfrac{1}{-\frac{1}{2} + 1}x^{-\frac{1}{2} + 1} + \text{ C}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \dfrac{1}{\frac{7}{2}} x^{\frac{7}{2}} \:-\:\dfrac{1}{\frac{1}{2}}x^{\frac{1}{2}} + \text{ C}\)
\(\int \dfrac{x^3 \:-\:1}{\sqrt{x}} \text{ dx} = \dfrac{2}{7} x^{\frac{7}{2}} \:-\:2x^{\frac{1}{2}} + \text{ C}\)
Ingat rumus pemfaktoran \(\color{blue} x^2\:-\:y^2 = (x + y)(x\:-\:y)\)
\(\int \dfrac{x^2\:-\:3^2}{x^2 + 3x} \text{ dx}\)
\(\int \dfrac{\cancel{(x + 3)}(x\:-\:3)}{x\cdot \cancel{(x + 3)}}\text{ dx}\)
\(\int \dfrac{x\:-\:3}{x}\text{ dx}= \int (1\:-\:\frac{3}{x}) \text{ dx}\)
Jadi, soal di atas dapat disederhanakan menjadi:
\(\int (1\:-\:\frac{3}{x}) \text{ dx}\)
Ingat \(\color{blue} \int \frac{1}{x} \text{ dx} = \ln |x| + \text{ C}\)
\(\int (1\:-\:3x^{-1} \text{ dx}= x \:-\:3\ln {|x|} + \text{ C}\)
Lakukan pembagian polinomial secara bersusun:
Hasil pembagian \((x^3\:-\:7x^2 + 13x\:-\:15)\) oleh \((x\:-\:5)\) adalah \((x^2\:-\:2x + 3)\)
Jadi, soal di atas dapat ditulis lebih sederhana
\(\int (x^2\:-\:2x + 3) \text{ dx}\)
\(\int (x^2\:-\:2x + 3) \text{ dx} = \dfrac{1}{2 + 1}x^{2 + 1}\:-\:\dfrac{2}{1 + 1} x^{1 + 1} + 3x + \text{ C}\)
\(\int (x^2\:-\:2x + 3) \text{ dx} = \dfrac{1}{3}x^{3}\:-\:\dfrac{2}{3} x^{2} + 3x + \text{ C}\)
